The book is a first introduction to a mathematical description of fields, flows and waves. It is designed for undergraduate students in mathematics, mathematical physics and engineering. It shows students how many mathematical topics are useful in constructing, analyzing and interpreting phenomena in the real world, in physics and in biology. It develops concepts of flux, conservation law and boundary value problems through simple examples of heat flow, electric potentials and gravitational fields. The ideas are developed through worked examples and a range of exercises (with solutions provided). The first seven chapters contain material for an introductory lecture course. It includes the continuum description, unsteady heat flow, fields and potentials, the Laplace and the Poisson equations, motion of an elastic string, fluid flow and elastic deformations. The next chapters are on waves in fluids, solids and electromagnetism, and on biomathematics. They show how an extension of earlier ideas leads to a description and explanation of important topics in modern technology and science.